Generalized Matrix Factorization

The text of DEMO program S.Funk

Generalized Matrix Factorization

There is no shortage of algorithms for matrix factorization, however, most of them do not provide support for hidden elements and for multiple elements in each matrix cell.

In general case we may not know all elements of the matrix and also have multiple elements for a particular cell. Let us look at the data below. It is sort of 3d matrix that has three approximately equal layers, so we have 3 elements for each row-column combination, but there are two row-column combinations, for which we do not have data at all. But we still want to find decomposition in a form of two matrices, that when multiplied produce some product close to each of these layers.

Layer 1
30	N/A	40	11
30	45	33	11
30	45	64	20
24	36	N/A	25
36	54	79	32

Layer 2
31	N/A	41	12
32	47	36	14
31	45	64	22
25	38	N/A	28
38	56	79	35

Layer 3
33	N/A	41	11
33	45	34	11
30	48	65	21
25	37	N/A	26
37	56	81	33

With suggested under link at the top algorithm we found cofactors and the product of them shown below.

Left cofactor
1.249	4.084	4.205
1.006	2.704	5.272
4.304	4.870	2.412
5.700	2.534	2.135
6.676	3.380	4.183

Right cofactor
1.444	2.293	7.864	4.063
2.830	4.494	5.069	-0.027
4.281	5.923	2.411	1.514

Product
31.363	46.127	40.666	11.331
31.671	45.683	34.328	11.997
30.320	46.039	64.345	21.002
24.539	37.101	62.816	26.318
37.110	55.271	79.715	33.362

The credit goes to someone under name S.Funk, who published this procedure several years ago for two dimensional matrices, and my role is reduced to generalization it into 3d matrices with approximately equal layers.

The impressive part of this algorithm is that (at the core) it needs only few lines of code.

Andrew Polar, Jan, 2017.